Algorithm 2: Fuzzy c-means clustering

1.

First consider n data points, $x=\left[\begin{array}{lllll}x\left(\text{1}\right)\hfill & x\left(\text{2}\right)\hfill & x\left(\text{3}\right)\hfill & \cdots \hfill & x\left(n\right)\hfill \end{array}\right]$ to be segregated into c clusters

2.

Take the initial value of center of clusters, c_k; where $k=\text{1},\text{2},\text{3},\cdots ,c$

3.

Evaluate grade (or degree) of membership u_ij i.e. the degree to which an x(i) belongs to the cluster with center c_j,

$u_{i,j}=\frac{1}{{\displaystyle {\sum }_{l=1}^c{\left(\frac{\left|x\left(i\right)-c_j\right|}{\left|x\left(i\right)-c_l\right|}\right)}^{\frac{2}{m-1}}}}$ ; for $i=1,2,3,\cdots ,n$ (4)

The entire vector is expressed at kth iteration as,

$U_{\text{1}}\left(k\right)=\left[u_{11},u_{21},u_{31},\cdots ,u_{n1}\right]$ , under cluster 1

$U_{\text{2}}\left(k\right)=\left[u_{12},u_{22},u_{32},\cdots ,u_n{}_2\right]$ , under cluster 2

$⋮$

$U_c\left(k\right)=\left[u_{1c},u_{2c},u_{3c},\cdots ,u_{nc}\right]$ , under cluster c

4.

Update the center c_j like,

$c_j=\frac{{\displaystyle {\sum }_{x\left(i\right)\in c_j}{u}_{ij}^{m}x\left(i\right)}}{{\displaystyle {\sum }_{x\left(i\right)\in c_j}{u}_{ij}^m}}$ (5)

5.

Repeat step 3 and 4 until $\left|U_j\left(k\right)-U_j\left(k+\text{1}\right)\right|<\epsilon$ , $j=\text{1},\text{2},\text{3},\cdots ,c$